Additional notes on the Leibniz Controversy

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to the world that he was the first inventor thereof. But for the future {illeg}{illeg} when ever he represents himself the first inventor of exponential equations he ought at the same time to acknowledge that Mr Newton gave him light into the invention by teaching him the use of dignities whose indices were fract surd {illeg}|or| indeterminate. [And whereas he pretends to have enlarged Geometry by Exponential Equations he ought to consider that the indices of all dignities are numbers & where the numbers are fluents, the æquations are Arithmetical & have no place in Geometry. Geometrical quantities may be represented by numbers but are not numbers & Arithmetic may be applyed to the resolution of Geometrical Problems: but such resolutions are only Arithmetical, & have no place in Geometry untill they are demonstrated by Geometrical Proposition & thereby become Geometrical Solutions. Nothing is legally admitted into Geometry\Geometry is composed of nothing else then/except\besides/ Definitions Axioms Lemmas & Propositions \Lemmas & Corollaries/ demonstrated synthetically, AndAnalysis is noth whether ancient or modern is only\only/ a mean of resolving Problemes: {illeg}\and/ the Resolutions ought to be turned into Geometrical Solutions by Compos\Geometrical/ Synthesis or Composition before the Problem Propositions be admitted into Geometry.]

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Mr Ne SrM|I|. Newton in his Letter of 24 Octob. 1676 wrote that he had two methods of resolving Invers Problemes of tangents & such like difficult ones, one of wch\methods/consisted in assuming a Series for any unknown quantity from wch the other unknown quantities might easily be conveniently be deduced & in the collection of the homologous terms of the resulting equation, for determining the terms of the assumed series. Mr Leibnitz has\s{illeg}|o|me years after/ published this method as his own claiming to himself the first invention thereof. If remains that he either renounce this claim publickly \& acknowledg that M|S|r I. Newton was the first Innovator/ or prove that he invented it before 24 Octob. 1676SrM|I|. Newton wrote the said Letter.

Sr I. Newton in his said Letter of 24 Octob. 1676 used\introduced into Analysis the use of/ fract surd & indefinite indices of dignities & in his letter of 24 Octob 1676 {illeg}proposed\represented/ to Mr Leibnitz th{illeg}|a|t {sic}{use}Problem of resolving\his methods extended to the resolution of/ affected æquations involving dignites|ie|s whose indices were fract or surd. Mr Leibnitz in his answer dated 21 Jun{e}1677 de mutually desired Mr newton to tell him what he thought of the resolution of æquations {illeg}|in|volving æquation dignities whose indices were surdindetermined quantities as Mr Newton calls fluents, such as are\as in the manner of the following examples such as were/\the æquations/xy+yx=x⁢y & xx+yy=x+y. And these æquations he calls exponentia{illeg}|l|, & has represented to

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|pag. 183. \l. 11/| In {illeg}|th|e year 1672 going to Paris he fell acquainted wth Mr Hugens & in the beginning of the next year came to London, & in February meeting Dr Pell at Mr Boyles.

Pag. 184. l. 1. In the end of February or beginning of March 1672/3 MrLeibnitz went from London to Paris \{illeg} carrying with him the Logarithmotechnia of Mercator of wch Dr Pell had given him \notice// & kept a correspondence with Mr Oldenburg about Arithmetical matters being hith till June{illeg}\following/ being hitherto unacquainted wth the higher Geometry. But the Horologium Oscillatorum of MrHugens being published in April, by the reading first of that book first & then of {illeg} Paschal's Letters & Gregory of st Vincent's Book de quadratura circuli &c {illeg}& by the i{illeg} he became acquainted with this Geometry, {illeg} & after some intermission of his correspondence wth Mr Oldenburg, wrote to him in July 1684 that he had a wonderful Theoreme, wch gave &c

Pag. 191. l. 1{illeg}|2|. He found out therefore this new Analysis after Aug. 27 1676. or rather after his being in England \wch was/ in October following.

Pag. 189. lin. 25. After the words — he wrote for Mr Newton's — add this section. After he began to study the higher geometry he fell upon h{illeg}|is|Triangulum characteristicum, And {illeg} & invented many particular Theorems like those of Gregory & Barrow, & went on to his Method founded on his Analytical Tables of Tangents & his Combinatory Art. And this was the top of his skill when he wrote his Letter of 27 Aug. 1676. For in that Letter he said of one part of this Method: Nihil est quod norim in tota Analysi momenti majoris. And of another part: Cujus vim ac potestatem nescio an quisquā hactenus sic consecutus. Ea vero nihil differt ab Analysi illa suprema ad {illeg} cujus intima Cartesius non pervenit. Est enim ad eam costituendam opus Alphabeto cogitationum humanarum.